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** This book has been updated by Jim Davies at the Oxford University Computing Laboratory and the new edition is available for download as a PDF file at the Using CSP website.
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** and has
** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
** Every surjective function has a right inverse.
** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.
** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.
** Antichain principle: Every partially ordered set has a maximal antichain.
** Every vector space has a basis.
** The closed unit ball of the dual of a normed vector space over the reals has an extreme point.
** Every field has an algebraic closure.
** Every field extension has a transcendence basis.
** The theorem that every Hilbert space has an orthonormal basis.
** Every Tychonoff space has a Stone – Čech compactification.
** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.
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** quadratic equation has degree two,
** cubic equation has degree three,
** quartic equation has degree four,
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